Aisles in derived categories

Transcription

1 Aisles in derived categories B. Keller D. Vossieck Bull. Soc. Math. Belg. 40 (1988), Summary The aim of the present paper is to demonstrate the usefulness of aisles for studying the tilting theory of D b (mod A), where A is a finitedimensional algebra. In section 1, we establish the equivalence of aisles with t-structures in the sense of [3] and give a characterization of aisles in molecular categories. Section 2 contains an application to the generalized tilting theory of hereditary algebras. Using aisles, we then give a geometrical proof of the theorem of Happel [7] which states that a finitedimensional algebra which shares its derived category with a Dynkinalgebra A can be transformed into A by a finite number of reflections. The techniques developed so far naturally lead to the classification of the tilting sets in D b (mod k A n ) presented in section 5. Finally, we consider the classification problem for aisles in D b (mod A), where A is a Dynkin-algebra. We reduce it to the classification of the silting sets in D b (mod A), which we carry out for = A n. We thank P. Gabriel for lectures on these topics and for his helpful criticisms during the preparation of the manuscript. Notations Let A be a finite-dimensional algebra over a field k. We denote by mod A the category of finitely generated right A-modules, proj A the full subcategory of mod A consisting of the projective A-modules,

2 C (proj A) the category of differential complexes over proj A which are bounded above, C b (proj A) the full subcategory of C (proj A) consisting of the complexes which are bounded above and below, C b (proj A) the full subcategory of C (proj A) consisting of the complexes X such that H i X = 0 for almost all i Z, H b (proj A) the homotopy category obtained from C b (proj A) by factoring out the ideal of the morphisms homotopic to zero [12], D b (A) := D b (mod A) the bounded derived category [12] of the abelian category mod A. 1 Aisles 1.1 Let T be a triangulated category with suspension functor S. A full additive subcategory U of T is called an aisle in T if a) SU U, b) U is stable under extensions, i.e. for each triangle X Y Z SX of T we have Y U whenever X, Z U, c) the inclusion U T admits a right adjoint T U, X X U. For each full subcategory V of T we denote by V (resp. V) the full additive subcategory consisting of the objects Y T satisfying Hom (X, Y ) = 0 (resp. Hom (Y, X) = 0) for all X V. The following proposition shows that the assignment U (U, SU ) is a bijection between the aisles U in T and the t-structures on T (in the sense of [3]). Proposition. A strictly (=closed under isomorphisms) full subcategory U of T is an aisle iff it satisfies a) and c ) c ) for each object X of T there is a triangle X U X X U SX U with X U U and X U U.

3 Proof. Suppose U satisfies a) and c ). The long exact sequence arising from the triangle in c ) shows that Hom (U, X U ) Hom (U, X) for each U U. If U X V SU is a triangle and U, V U then Hom (X,?) vanishes on U and X U. In particular, the morphism X U SX U of c ) has a retraction, hence X U = 0 and X = X U lies in U. Conversely, let U satisfy a),b),c). According to b), U is strictly full. In order to prove c), we form a triangle X U ϕ ψ ε X Y SX U over the adjunction morphism ϕ. Let V U and f Hom (V, Y ). We insert f into a morphism of triangles X U h W V εf SX U g f X U ϕ X ψ Y ε SX U According to b), W lies in U. By assumption, g factors uniquely through ϕ. Therefore, h has a retraction and εf = 0. So f factors through ψ and even through ψϕ = 0 since V U. 1.2 For certain triangulated categories, condition 1.1c) can still be weakened. We call an additive category T a molecular category if each object of T is a finite direct sum of objects with local endomorphism rings. In particular, if A is a finite-dimensional algebra over a field k, the category D b (A) is a molecular category: For all X, Y D b (A), we have dim k Hom (X, Y ) < and for each indecomposable U of D b (A), End (U) is local since End C (proj (V ) is local for each indecomposable V of A) C (proj A). If T is a molecular category, we denote by ind T a full subcategory of T whose objects form a system of representatives for the isomorphism classes of indecomposables of T. 1.3 Proposition. Let T be a triangulated molecular category and U a full additive subcategory which is closed under taking direct summands. The subcategory U is an aisle in T iff it satisfies a), b) and c ). c ) For each object X of T the functor Hom (?, X) U is finitely generated, i.e. there is U U and an epimorphism Hom U (?, U) Hom T (?, X) U.

4 Proof. The claim follows from the Lemma. Let S be a suspended [9] molecular category and F : S op Ab a cohomological functor, i.e. for each triangle X u Y v Z w SX of S the sequence F X F u F Y F v F Z is exact. The functor F is representable iff it is finitely generated. Proof. Let F be finitely generated. Because S is a molecular category F has a projective cover Hom S (?, X) ϕ F in the abelian category of the additive functors S op Ab. We shall show that ϕ is a monomorphism. Let Y S and f Hom (Y, X) such that ϕ Hom (?, f) = 0. We form a triangle Y f X g Z h SY in S. By the Yoneda Lemma we conclude from the exactness of F Y F f F X F g F Z that ϕ factors through Hom (?, g). By construction of ϕ, Hom (?, g) admits a retraction. Hence g admits a retraction and f = Proposition. Let U, V be aisles in a triangulated category T such that V U. Then W := U V = {X T : There is a triangle U X V SU with U U,V V} is also an aisle in T (cf. [3, 1.4]) Proof. We shall verify the conditions of Proposition 1.1. Only c ) is not immediate from the definitions. Let X T. We form a diagram Y (X U ) V X U = X U V X U X where triangles are marked by and tailed arrows denote morphisms of degree 1. By definition, Y W, X U V V. Since X U V is

5 an extension of S((X U ) V ) by X U it also lies in U, hence in W = U V. We obtain the required triangle Y X X U V SY by forming an octahedron with base X X U X U V. 2 Tilting sets 2.1 Let A be a finite-dimensional algebra over a field k. We consider the problem of determining all finite-dimensional k-algebras B such that D b (B) is S-equivalent [9] to D b (A). A tilting set (cf. [7]) in D b (A) is a finite subset {T 1,..., T s } ind D b (A) such that Hom (S l T i, T j ) = 0 for all i, j and all integers l 0. Each fully faithful S-functor F : D b (B) D b (A) gives rise to the tilting set {T ind D b (A) : T = F P for some indecomposable projective B-module P }. Conversely, let {T 1,..., T s } be a tilting set and B := End ( s i=1 T i ). By [9, 3.2], the obvious functor from proj B to D b (A) extends to a fully faithful S-functor E : H b (proj B) D b (A) (put E := Cb (proj A) in [9, 3.2]). We make the additional assumptions a) Hom (T i, T j ) = 0 i > j and Hom (T i, T i ) is a skew field i. b) gldima <. Assumption a) implies gldimb <. By composing E with a quasiinverse of the equivalence H b (proj B) D b (B) we obtain a fully faithful S-functor F : D b (B) D b (A). Proposition. The essential image of F is an aisle in D b (A). In particular, F has a right adjoint. Proof. Let U i be the strictly full triangulated subcategory of D b (A) generated by T i. Since D b (End (T i )) U i, assumption b) and proposition 1.3 imply that U i is an aisle in D b (A). The essential image of F equals U s U s 1 U 2 U 1 (by [3, ], is associative). The claim now follows from proposition Theorem. Let A be a hereditary finite-dimensional k-algebra and {T 1,..., T s } a tilting set in D b (A). The functor F is an S-equivalence iff s equals the number of isomorphism classes of simple A-modules.

6 Proof. By [7, 7.3] assumption a) is satisfied for an appropriate numbering of the T i and, since A is hereditary, so is assumption b). An S-equivalence D b (B) D b (A) induces an isomorphism of the Grothendieck-groups K 0 (B) K 0 (A). Therefore, since s equals the number of isomorphism classes of simple B-modules, the condition is necessary. For the converse, it is enough to show that {F X : X D b (B)} = 0, by proposition 2.1. Let Y D b (A) be indecomposable and such that Hom (F X, Y ) = 0, X D b (B). This implies < [F X], [Y ] >= 0, where [.] denotes the canonical map D b (A) K 0 (A) and <.,. > denotes the canonical bilinear form on K 0 (A) [7]. F induces a section K 0 (B) K 0 (A) (the right adjoint of F yields a retraction). Since rankk 0 (B) = s = rank K 0 (A), we have K 0 (A) = {[F X] : X D b (B)}, hence [Y ] = 0 and, since A is hereditary and Y is indecomposable, Y = 0 [7]. 3 Dynkin-algebras 3.1 Let B be a basic, connected, finite-dimensional algebra over an algebraically closed field k. Assume that there exists a simple, projective, non-injective right B-module P. Define Q by B = Q P. Then T = τ P Q is a tilting module in mod B (cf. [5], [8]), where τ P denotes a preimage of P under the Auslander-Reiten-translation. The derived functors F = RHom B (T,?) : D b (B) D b (B P ) and G = L(? BP T ) are quasi-inverse S-equivalences, where B P = End (T B ) is obtained from B by reflection in P [4]. Thus, the derived category D b (B) is invariant under reflections. Conversely, we have the Theorem. (Happel) Let D b (B) be S-equivalent to D b (A) where A is a Dynkin-algebra (=path algebra [6, 4.1] of a Dynkin quiver). Then A is obtained from B by a finite number of reflections. Proof. Let U = {X D b (B) : H i X = 0 i > 0} be the natural aisle in D b (B) [3, 1.3.1], V the natural aisle in D b (A) and E : D b (B) D b (A) an S-equivalence with [V] [EU], where [?] denotes the set of isomorphism classes of indecomposables in?. If [V] = [EU], E induces an equivalence

7 of the hearts (=hearts of the corresponding t-structures) of U and V, i.e. of mod B and mod A. In general, [V] is obtained from [EU] by the omission of finitely many isomorphism classes, as it is apparent from Happel s description of ind D b (A) [7]. By induction, we are reduced to the case [V] = [EU]\{EM}, where M is a source of ind U, i.e. Hom (V, M) = 0 V ind U, V M and S M U. The sources of ind U are isomorphic to the simple projectives of mod B. It is therefore enough to prove the 3.2 Lemma. In the setting of 3.1 let U be the natural aisle in D b (B), W the natural aisle in D b (B P ) and W its essential image under G. Then W = {X U : P is not a direct summand of X}. Proof. Let U P = {X U : P is not a direct summand of X}. It follows from pdim BP T 1 that GW SU hence SU W. We also have SU U P and U P = (SU) (U P mod B), W = (SU) (W mod B). By [2], U P mod B is just the torsion theory generated by T, which by [5] coincides with the essential image of mod B P under H 0 G =? BP T. 4 Complete tilting sets in D b (k A n ) Let A be a finite-dimensional algebra over a field k. We define the spectrum of a tilting set M in D b (A) as the full subcategory of D b (A) whose objects are the elements of M. We call a tilting set in D b (A) complete if its cardinality equals the number of isomorphism classes of simple A- modules. Let A be the path algebra of the quiver A n : n [6]. We identify [7] the category ind D b (A) with the mesh-category k(z A n ) [11, 2.1] of the translation quiver Z A n : (1,n) (0,n) (1,n 1) (0,n 1) (1,3)... (1,2) (0,2) (1,1) (0,1)

8 By an A n -quiver we mean an oriented tree K having n vertices and whose set of arrows is decomposed into a class of α-arrows and a class of β-arrows such that in each vertex of K there terminate at most one α- arrow and one β-arrow and there originate at most one α-arrow and one β-arrow. Now let K be an A n -quiver. For each vertex x of K let x α (resp. x α ) be the number of vertices y of K such that the shortest walk from y to x ends (resp. begins) with an α-arrow terminating (resp. originating) in x. Analogously, we define x β and x β. Then there is exactly one map of the underlying sets of vertices K 0 (Z A n ) 0, x (gx, hx) such that a) min x K gx = 0, b) (gy, hy) = (gx, hx + x β + y β + 1) for each α-arrow x α y and c) (gy, hy) = (gx + x α + y α + 1, hx x α y α 1) for each β-arrow x β y. Because hx = 1 + x α + x β, this map is indeed well defined. Let M K denote its image. Theorem. The assignment K M K induces a bijection from the isomorphism classes of A n -quivers to the complete tilting sets M in ind D b (k A n ) = k(z A n ) with min (g,h) M g = 0. Moreover, the spectrum of M K is described by the quiver K bound by all possible relations αβ = 0 and βα = 0 (cf. [1]). Proof. Let K be an A n -tree. We call x K 0 a knot of K if x is contained in a full subtree of one of the forms α x α, α x β, β x α, β x β. The other vertices of K are called peaks. We term (g, h) (Z A n ) 0 a marginal vertex if h = 1 or h = n and we call the other vertices of Z A n inner vertices. We use induction on the number m of knots of K. If m = 0, K has one of the forms K α = α β α... or K β = β α β...

9 X 1 X 2 X r P SP Y S s P Y 2 Y 1 Figure 1: Z A n with given vertex P It is easy to see that the corresponding sets M Kα, M Kβ are exactly the complete tilting sets M of ind D b (ka n ) which only consist of marginal vertices and satisfy min (g,h) M g = 0. Now let m > 0. We first describe the complete tilting sets in k(za n ) which contain a given inner vertex P. The tilting sets {X 1,..., X r, P } and {Y 1,..., Y s, P } (cf. Fig. 1) give rise to fully faithful embeddings j X : D b (ka r+1 ) D b (ka n ) and j Y : D b (ka s+1 ) D b (ka n ). We may assume that j X (ZA r+1 ) 0 (ZA n ) 0 and j Y (ZA s+1 ) 0 (ZA n ) 0, in particular j X P X = P and j Y P Y = P where P X = (0, r + 1) (ZA r+1 ) 0 and P Y = (0, s + 1) (ZA s+1 ) 0. With these notations, the complete tilting sets M in k(za n ) containing P are exactly the sets j X (L) j Y (N) where L (ZA r+1 ) 0 and N (ZA s+1 ) 0 are complete tilting sets containing P X and P Y, respectively. Here, the set of marginal points of M equals {j X (R) : R is a marginal point of L \ P X } {j Y (R) : R is a marginal point of N \ P Y }. For the corresponding spectra we have the pushout diagram k i X L i Y j X N j Y M where k is considered as a category with one object and i X and i Y are fully faithful with i X (k) = P X and i Y (k) = P Y. Combined with the induction hypothesis, this description shows that the spectra of the complete tilting sets in k(z A n ) are described by the A n -quivers with all relations αβ = 0 = βα and that peaks are mapped to marginal points by the

10 corresponding isomorphisms of categories. So let M be a complete tilting set in k(z A n ) whose spectrum is described by K. We claim that the corresponding bijection e : K 0 M (Z A n ) 0 satisfies conditions b) and c). This is obvious if x, y are peaks of K since then ex, ey are marginal points. If for example x is a knot we apply the above construction with P = ex and the claim follows from the induction hypothesis. 5 Aisles in D b (k ) 5.1 Let U be an aisle in a triangulated category T. The heart of U is the full subcategory U 0 = U SU of T ; the associated cohomology functor HU 0 : T U 0 is given by X (X U ) SU [3, ]. The aisle U is faithful if the inclusion U 0 T extends to an S-equivalence D b (U 0 ) n N S n U; it is separated if n N S n U = 0. Let k be the path algebra [6] of a Dynkin-quiver. For each subset M ind D b (k ) let F(M) be the smallest strictly full subcategory of D b (k ) which contains M and is stable under S and closed under extensions and direct summands. By proposition 1.3, F(M) is an aisle. The assignment {T 1,..., T s } F(T 1,..., T s ) is a bijection between the tilting sets in D b (k ) and the faithful aisles. We shall generalize the concept of a tilting set in order to obtain an analogous description of all separated aisles in D b (k ). A silting set in D b (k ) is a finite subset {R 1,..., R s } ind D b (k ) such that Hom (R i, S l R j ) = 0 i, j and l > 0. Theorem. a) The assignment {R 1,..., R s } F(R 1,..., R s ) is a bijection between the silting sets in D b (k ) and the separated aisles in D b (k ). If {R 1,..., R s } is a silting set and W = F(R 1,..., R s ) we have b) W (SW) ind D b (k ) = {R 1,..., R s } c) s 0, and s = 0 n N S n W = D b (k ) d) W = {X n N S n W : Hom (R i, S l X) = 0 i, l > 0}

11 SQ S 2 Q Q Figure 2: Example of V ( ) and U ( ) in Z A n e) HWR 0 1,..., HWR 0 s form a system of representatives of the isomorphism classes of indecomposable projectives of W 0, and HW 0 induces an equivalence between the full subcategories {R 1,..., R s } and {HWR 0 1,..., HWR 0 s } of D b (k ). Proof. 1st step: The following variant of a construction by Parthasarathy [10] allows us to use induction on 0. Let W be a separated aisle in D b (k ) and Q a source (3.1) of ind W. We may assume that admits a unique source q and that Q = (0, q). Let Q be the full subcategory of D b (k ) whose objects are the direct sums of objects S n Q, n Z. The tilting set {(0, r) : r 0, r q} k(z ) yields a fully faithful S-functor D b (k ) D b (k ) (2.1), where is obtained from by omitting the vertex q and all arrows originating in q. The essential image of this S-functor equals Q. We claim that W = U V, where U = W Q and V = W Q. Obviously, we have W U V. Conversely, let X be an indecomposable in W which is not isomorphic to Q. We have the triangle X Q X X Q SX Q (1.3). The assumptions on X imply that X Q lies in SV. Thus, as an extension of X by S X Q V W, X Q lies in W Q = U. 2nd step: b) Let the numbering be chosen in such a way that Hom (R j, R i ) = 0 i < j. We apply the construction of the first step to the source Q = R 1 of ind W. Let X W (SW) be indecomposable. We have the triangle X Q X X Q SX Q. As in the first step, either X = R 1 or X Q SV and in this case we infer X Q = 0 and X U (SU). The

12 claim now follows from the induction hypothesis. a) Because of b) we only have to show surjectivity. In the setting of the first step let R 1 = Q. We complete R 1 to a system of representatives {R 1,..., R s } of the indecomposables of W (SW). Then {R 2,..., R s } is a system of representatives of the indecomposables of U (SU). According to the induction hypothesis, we have U = F(R 2,..., R s ), and therefore W = U V = F(R 1,..., R s ). The proof of c) is left to the reader. d) Obviously, W is contained in the aisle given in the assertion. Conversely, let X = S n Y (Y W, n N) and Hom (R i, S l X) = 0 i, l > 0. By induction we conclude S n Y U U. Using Hom (R 1, SU) = 0 and the triangle Y U = Y Q Y Y Q SY Q we infer Hom (S l R 1, S n Y Q ) = 0 for all l > 0 and S n Y Q V. e) From Hom (R 1, SW) = 0 it follows that R 1 = H 0 W R 1 is projective in W 0. The rest of the assertion follows from the Lemma. Let U, V be aisles in a triangulated category T U V and let W = U V. (cf. proposition 1.4) such that a) V 0 W 0 and H 0 V W 0 is right adjoint to this inclusion. b) H 0 U W 0 is exact and H 0 W U 0 is left adjoint to H 0 U W 0 and fully faithful. c) For each A W 0 we have an exact sequence H 0 WH 0 UA A H 0 WA U 0 d) H 0 W W (SW) is fully faithful and for X U we have H 0 WX = H 0 WH 0 UX. We leave the proof of the lemma to the reader (compare with [3]). 5.2 In the setting of 5.1 let {T 1,..., T s } be a tilting set in D b (k ). Suppose that the numbering has been chosen in such a way that Hom (T j, T i ) = 0 j > i. Let p : {1,..., s} N be a non-decreasing function with p(1) = 0.

13 Proposition. a) R 1 := S p (1) T 1,..., R s := S p (s) T s form a silting set in D b (k ). b) With U = F(T 1,..., T s ), W = F(R 1,..., R s ) we have for each i Z U 0 S i W = {X U 0 : Hom (T j, X) = 0 for each j with i < p(j)} W = {X U : H i UX U 0 S i W, i Z} c) The full subcategory {R 1,..., R s } of D b (k ) is isomorphic to the disjoint sum of the full subcategories C j = {T i : p(i) = j}, j N, of {T 1,..., T s }. We leave the proof to the reader. 5.3 Theorem. Each silting set in D b (k A n ) (cf. section 4) is of the form given in 5.2 Proof. Let {R 1,..., R s } be a silting set in D b (ka n ). 1st step: {R 1,..., R s } is contained in a complete silting set (= silting set of maximal cardinality). If s < n we have T 0 (5.1 c), where T = m N S m F(R 1,..., R s ). Because gldimk < we can find an indecomposable R 0 T such that Hom (R 0, S l R i ) = 0 for all l > 0, i = 1,..., s. Then {R 0,... R s } is a silting set. The claim now follows by induction on n s. 2nd step: By the first step we may assume n = s. Let the numbering of the R i be non-decreasing with respect to the order on ind D b (ka n ) generated by the arrows of ZA n. With the notations of the first step of the proof of theorem 5.1 we set Q = R 1. The induction hypothesis applied to {R 2,..., R n } Q yields a tilting set {T 2,..., T n } which corresponds to a complete tilting set in D b (k ). Therefore the connected components of the spectrum of {T 2,..., T n } are precisely the intersections of {T 2,..., T n } with the connected components of ind Q.

14 Figure 3: A silting set in Z D 4 Let C be a connected component of ind Q. Because Hom (Q,?) C = 0 and {T 2,..., T n } C is a complete tilting set in C, Hom (Q,?) does not vanish on S l {T 2,..., T n } C for some l Z. Hence we may assume that {Q, T 2,..., T n } is connected. Because {X C : Hom (Q, X) 0} is linearly ordered by X X : Hom (X, X ) 0 (cf. figure 2), {Q, T 2,..., T n } must be a tilting set in D b (ka n ). Setting T 1 = Q we have R i = S p (i) T i for some function p : {1,..., n} Z with p(1) = 0. It is easy to see that Hom (T i, T j ) 0 implies p(i) p(j). 5.4 Remarks: a) Silting sets can always be completed (cf. the first step of the above proof) but in general, tilting sets cannot : {(0, 1), (3, 3)} Z A 3. b) The silting set of figure 3 is not of the form given in 5.2 since 5.2 c) cannot be satisfied. References [1] I. Assem, D. Happel, Generalized tilted algebras of type A n, Comm. Alg. 9 (1981), ; Erratum, Comm. Alg. 10 (1982), 1475 [2] M. Auslander, M. I. Platzeck, I. Reiten, Coxeter functors without diagrams, Trans. Amer. Math. Soc. 250 (1979), 1-46 [3] A. A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Astérisque 100 (1982) [4] I. N. Bernstein, I. M. Gelfand, V. A. Ponomarev, Coxeter functors and Gabriel s theorem, Uspekhi Mat. Nauk 28 (1973); translated in Russian Math. Surveys 28 (1973), 17-32

15 [5] K. Bongartz, Tilted algebras, Springer LNM 903 (1982), [6] P. Gabriel, Auslander-Reiten sequences and representation-finite algebras, Springer LNM 831 (1980), 1-71 [7] D. Happel, On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), [8] D. Happel, C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274(2) (1982), [9] B. Keller, D. Vossieck, Sous les catégories dérivées, C. R. Acad. Sci. Paris 305 (1987), [10] R. Parthasarathy, t-structures dans la catégorie dérivée associée aux représentations d un carquois, C. R. Acad. Sci. Paris 304 (1987), [11] C. Riedtmann, Algebren, Darstellungsköcher, Überlagerungen und zurück, Comment. Math. Helv. 55 (1980), [12] J.-L. Verdier, Catégories dérivées, état 0, SGA 4 1/2, Springer LNM 569 (1977), B.K. : Mathematik, G 28.2, ETH-Zentrum, 8092 Zürich, Switzerland; D.V. : Mathematisches Institut, Universität Zürich, Rämistrasse 74, 8051 Zürich, Switzerland